lab #3: operational amplifiers - department of physics |...

P309 Intermediate Lab, Indiana University Dept. of Physics

Last revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15

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Lab #3: Operational Amplifiers Goal: So far we have looked at passive circuits composed of resistors, capacitors and inductors. The problem with passive circuits is that the real part of the impedance always decreases the amplitude of voltage and current in the circuit. Often we wish to take a small voltage or current and amplify it, so that we can measure it with greater precision. We might also want to add, subtract, integrate or differentiate two or more voltage or current amplitudes. Amplifiers allow us to perform all of these linear mathematical operations and more on an AC or DC voltage or current. The operational amplifier (op-amp) is a type of integrated circuit amplifier with properties that makes implementing these functions particularly simple. In this laboratory, you will learn the basic properties of an ideal op-amp, how to use operational amplifiers with various types of feedback control to perform simple transformations of an input signal and also some of the limitations of real op-amps. You will also apply the integrator circuit to measure the amplitude and direction of earth’s magnetic field in the laboratory. For a good primer on op-amps, see Wikipedia (https://en.wikipedia.org/wiki/Operational_amplifier). Equipment: OP07 op-amp, proto-board, assorted resistors and capacitors, DMM, oscilloscope, large inductor coil.

1 Introduction: A classical amplifier has two inputs: a ‘non-inverting’ input labeled “+,” and an ‘inverting’ input labeled “–.” Call the voltage at the “+” input +𝑉𝑉 and at the “−“ input −𝑉𝑉. The open-loop voltage output of the output of amplifier is: 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 = 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 × (+𝑉𝑉 − −𝑉𝑉). (eq. 1) For a normal amplifier, like a stereo amplifier, 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 is adjustable and we operate the amplifier with the output completely separate from the inputs. Operational amplifiers have a very high gain, 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺~106, which is not too useful in an open-loop configuration, unless you are looking at an input voltage in the micro-Volt range. Indeed, in an ideal op-amp, we assume that 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺~∞, in which case, 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜~ ± ∞, unless +𝑉𝑉= −𝑉𝑉. Negative feedback between output and input (i.e. where a bigger 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 reduces 𝑉𝑉𝑖𝑖𝑖𝑖) allows many practical op-amp applications, where the amplifier has linear response over more conditions than an open-loop amplifier (e.g. we can design the feedback so that the gain does not change despite changes in temperature). In most useful op-amp circuits, we determine the negative feedback by connecting the output of the op-amp to one or both inputs via appropriate passive components (resistors, capacitors, inductors,…). Figure 2 shows the simplest such

Figure 1: Amplifier in open-circuit mode, showing +, − and 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 connections.

https://en.wikipedia.org/wiki/Operational_amplifier



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configuration. As in all stable circuits using op-amps, the amplifier will set 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 to be whatever is necessary to make +𝑉𝑉= −𝑉𝑉. The arrangement of the feedback determines the function of the op-amp circuit. Negative feedback is an important and somewhat counter-intuitive concept. Please review it at: https://en.wikipedia.org/wiki/Negative_feedback

We can determine the function of an ideal op-amp circuit from two ‘golden’ rules: • No current flows in or out of either of the two inputs to the op-amp. • 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 in any negative-feedback configuration strives to make the voltage difference

between the two inputs zero, i.e., +𝑉𝑉= −𝑉𝑉. Our op-amp is an OP07, an integrated circuit with dozens of transistors, packaged in an 8-pin plastic DIP (Dual In-Line Package). You will find a data sheet for the OP07 at the end of this document. Unlike the other components you have studied so far, the op-amp is an active device: it requires a power supply to operate. The OP07 op-amp requires power-supply voltages of ±15 V. If the output wants to exceed the supply voltage, the signal is ‘clipped,’ i.e., if equation 1 predicts 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 > 15V, then the actual 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 = 15V, and if equation 1 predicts 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 < −15V, then the actual 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 = −15V. Clipping is one of the differences between a real and an ideal op-amp. Question: What is the open-loop gain of the OP07 op-amp (look at the data sheet at the end of this write-up)?

2 Inverting Amplifier We will first build a circuit to multiply the input signal by a fixed negative 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺. Follow Figure 2 to build this circuit. In this op amp configuration, connect the input signal through the series input resistor R1 to the inverting input ‘−‘ and also connect the feedback resistor R2 to the inverting input ‘−‘. Connect the non-inverting input ‘+’ to ground. The op-amp gain is given by

Figure 2. Inverting amplifier circuit. The figure shows the two power supply pins to the op-amp, 𝑉𝑉+ and 𝑉𝑉−. Most op-amp schematics do not show these pins, but you always must connect the power supply to these pins for the op-amp to function. Remember that the + and − pins are not the same as the 𝑉𝑉+ and 𝑉𝑉− power supply pins.

-

+

_

R

V+

V-

DIP, top view

V

R

V = - V R / R

1

out 12inin

2

Function Generator 𝑉𝑉𝑖𝑖𝑖𝑖~

Oscilloscope

Channel 2 Channel 1

𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜

https://en.wikipedia.org/wiki/Negative_feedback



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𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 =1

2

RR

VV

i n

o u t −= . (eq. 2)

Question: derive equation 2 for this circuit starting with the two golden rules. Using the Proto Board, build the inverting amplifier as shown in Figure 2. Pick R1 and R2 to have nominal resistances of 1kΩ and 10kΩ so that 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺~ − 10. Use a DMM to measure the actual resistance of the resistors and calculate the expected value for 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺. Refer to the photo in Figure 10 to see what your configuration will look like. Use a simple color scheme to help you remember the function of the different wires on the breadboard; e.g., red for power, green for ground, white or blue for signals. Use a signal generator to produce a 1kHz sine wave of 1V peak-to-peak amplitude with no DC offset for 𝑉𝑉𝑖𝑖𝑖𝑖. Use the oscilloscope to measure 𝑉𝑉𝑖𝑖𝑖𝑖 and 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 simultaneously. Determine the gain 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 = 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜

𝑉𝑉𝑖𝑖𝑖𝑖 .

Questions: Compare your measured 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 to the theoretical value 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝑜𝑜ℎ𝑒𝑒𝑜𝑜𝑒𝑒𝑒𝑒𝑜𝑜𝑖𝑖𝑒𝑒𝑒𝑒𝑒𝑒 = −𝑅𝑅2𝑅𝑅1

. Change the frequency of the function generator to 100Hz and 10kHz and measure the gain again. Is the gain independent of frequency? Change the input peak-to-peak voltage to 0.1V, 0.2V, 0.5V and 1.5V. To get a small voltage on the function generator, pull out the amplitude knob, which reduces the voltage by a factor of 10. Is 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 independent of the input voltage (i.e. is the amplifier linear)? Clipping Increase the signal generator amplitude until you observe clipping of 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜. At what output voltage do you see clipping? Change the power supply voltages to the op-amp (first 𝑉𝑉+, then 𝑉𝑉−. What happens to the output? Sketch what you observe and label the graph of 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 vs. 𝑡𝑡 with respect to 𝑉𝑉+ and 𝑉𝑉−. Slew Rate An ideal op-amp has an output voltage that changes instantly as the input voltage changes. A real op-amp has a maximum change in output voltage/second called the slew rate. Estimate the slew rate of your op-amp by setting the function generator to produce a square wave signal. Display both the square wave input voltage and the output voltage on the oscilloscope. Increase the frequency of the signal until the shapes of the waves in the two traces are clearly different. Now sketch or record the traces and measure the maximum 𝑑𝑑𝑉𝑉

𝑑𝑑𝑜𝑜

for the op-amp. Compare this result to the slew-rate quoted in the data sheet for the op-amp. Question: How can the finite slew rate of an op-amp affect its function? You should notice that once 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 is limited by the slew rate, the output voltage is no longer proportional to the input voltage and the shape of the output waveform is no longer the same as the shape of the input waveform. Describe what happens instead? Suppose you connect a sine-wave input signal to the op-amp of a fixed peak-to-peak amplitude, 𝑉𝑉𝑖𝑖𝑖𝑖 = 𝑉𝑉𝑝𝑝−𝑝𝑝

2sin (𝜔𝜔𝑡𝑡). If you

increase the frequency, the output signal will change from a sine wave to a triangle wave. Why? Calculate the theoretical 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 of the op-amp circuit as a function of the 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺, the slew rate, 𝑉𝑉𝑝𝑝−𝑝𝑝 and 𝜔𝜔. You should find that for high frequencies the op-amp can only



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amplify small amplitude signals and for large amplitudes it can only amplify lower frequencies. Derive the relationship between the maximum amplitude and maximum frequency at which the op-amp linearly amplifies the input signal. Now, repeat your experiment with a sine-wave input for three different 𝑉𝑉𝑝𝑝−𝑝𝑝 �𝑉𝑉𝑝𝑝−𝑝𝑝

5,𝑉𝑉𝑝𝑝−𝑝𝑝 𝐺𝐺𝐺𝐺𝑎𝑎 5 𝑉𝑉𝑝𝑝−𝑝𝑝�. For

each 𝑉𝑉𝑝𝑝−𝑝𝑝 sweep the frequency in powers of 100 and measure the output peak-to-peak voltage and the wave shape. Compare your results to your theoretical calculation.

Offset Voltage Connect the circuit shown in Figure 3. For an ideal op-amp, 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 = 0V if +𝑉𝑉= −𝑉𝑉. A real op-amp, will have 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 = a small offset voltage 𝑉𝑉𝑂𝑂𝑂𝑂, when +𝑉𝑉= −𝑉𝑉. Measure the offset voltage of the OP07. Use the circuit in Figure 3, and change R1 and R2 to have nominal resistances of 10Ω and 10kΩ so that 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺~ − 1000. As usual, measure both 𝑅𝑅1 and 𝑅𝑅2 to calculate 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝑜𝑜ℎ𝑒𝑒𝑜𝑜𝑒𝑒𝑒𝑒𝑜𝑜𝑖𝑖𝑒𝑒𝑒𝑒𝑒𝑒. Set 𝑉𝑉𝑖𝑖𝑖𝑖 = 0V by connecting the input of the resistor to ground. Now measure 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜with the oscilloscope and also with a DMM. Question: Consider R1 and R2 as a voltage divider. What is 𝑉𝑉−? Compare the measured offset voltage with 𝑉𝑉𝑂𝑂𝑂𝑂 specified in the OP07 data sheet.

3 Non-inverting Amplifier What if we don’t want to have the output voltage inverted with respect to the input voltage? Consider the non-inverting linear amplifier circuit in Figure 4. Here the input voltage connects to the non-inverting input and the voltage divider returns a fraction of the output voltage to the inverting input. Use the same resistors that you used in Section 2 for a nominal 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺~ − 10 to construct the circuit. Measure Vin and Vout, determine the actual gain. Question: Using the golden rules for op-amps show that the theoretical value for the gain of this circuit is:

Figure 3. Measurement of offset voltage by grounding the input voltage. You will need to use 𝑅𝑅1 = 10Ω,𝑅𝑅2 = 10kΩ. Set the trigger mode of the oscilloscope to “Line” so you can measure the DC offset voltage. Remember to connect the power supply to the 𝑉𝑉+ and 𝑉𝑉− power supply pins.

-

+

_

R

V+

V-

DIP, top view

V

R

V = - V R / R

1

out 12inin

2

Oscilloscope Channel 2 and DMM




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𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 =𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜𝑉𝑉𝑖𝑖𝑖𝑖

= 1 +𝑅𝑅2𝑅𝑅1

. (eq. 3)

Compare your experimental and theoretical results. Change the frequency of the function generator to 100Hz and 10kHz and measure the gain again. Is the gain independent of frequency? Change the input peak-to-peak voltage to 0.1V, 0.2V, 0.5V and 1.5V. Is 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺

independent of the input voltage (i.e. is the amplifier linear)?

4 Integrator

Op-amps can be used to construct a circuit that integrates an electrical signal over time (Figure 5). A capacitor serves as the memory of the integrator. To clear the memory, we simply short circuit the capacitor by closing a switch. When we open the switch, the integration starts (𝑡𝑡 = 0). Question: Use the two golden rules, to show that for a time-dependent input voltage,

𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜(𝑡𝑡) = −1𝑅𝑅𝑅𝑅

�𝑉𝑉𝑖𝑖𝑖𝑖(𝑡𝑡′)𝑎𝑎𝑡𝑡′.𝑜𝑜

0

(eq. 4)

Figure 4. Non-inverting amplifier circuit. The figure does not show the two power supply pins to the op-amp, 𝑉𝑉+ and 𝑉𝑉−, but you always must connect the power supply to these pins for the op-amp to function. Note that the wire to 𝑅𝑅1 does not connect to the wire from 𝑉𝑉𝑖𝑖𝑖𝑖. Connect Channel 1 of the oscilloscope to the Function generator directly as in Figure 2.

-

+

_

V

R

R

V = V (1 + R / R )out in 12

1

2

in

𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑉𝑉𝑖𝑖𝑖𝑖 �1 +𝑅𝑅2𝑅𝑅1�

No connection here

Function Generator

Oscilloscope

Channel 2 Channel 1




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Drift First reset the integrator by briefly pressing the switch on the 2µF capacitor. Connect the input of the resistor to ground. Since the voltage on the “−“ input of the op-amp is 0V, 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜should remain zero for an ideal op-amp. Usually, however, the output will drift because the golden rules are not exactly true. Measure the drift rate in Volts/second from your oscilloscope trace.

-

+

_

VV

inout

R

C

pin 1 pin 8

Figure 5. Basic voltage integrator circuit. Remember to connect the power supply to the op-amp. Set the oscilloscope to a very slow scan time and use the “Run/Stop” button to make it scan slowly across the screen.

Switch

Oscilloscope Channel 2

-

+

_

VV

inout

R

C

pin 1 pin 8

20k pot.

+V

Figure 6. Voltage integrator circuit with drift control. Attach a blue precision 20kΩ potentiometer connected to the +15V power supply to pins 1 and 8 of the op-amp. Remember to connect the power supply to the op-amp. Set the oscilloscope to a very slow scan time and use the “Run/Stop” button to make it scan slowly across the screen.

Switch




7

To reduce this drift, the OP07 provides an offset trim that allows you to adjust the balance of the two inputs. Build the circuit in Figure 6, by installing the offset trim, connecting a blue precision 20kΩ potentiometer (variable resistor) between pins 1 and 8 of the op-amp. Connect the adjustable contact of the potentiometer to the +15V supply. Adjust the potentiometer until the drift of the integrator is as near zero as possible. Use the white adjusting tool (a miniature screwdriver) to rotate the potentiometer. Determine the residual drift rate in Volts/second (you will need this result in Section 5). To show that the circuit integrates the input voltage as in equation 3, build the circuit in Figure 7 and apply a constant voltage 𝑉𝑉0 to the input. In this case, equation 3 tells us that the output voltage is a linear function of the time. Use the 10kΩ potentiometer on the Proto-Board to make a voltage divider to generate a small 𝑉𝑉0~10mV, so 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 takes about 30s to increase from 0V to 15V. Select the divider resistors accordingly.

Question: Why should 𝑅𝑅0 be less than 𝑅𝑅? Measure the rate of increase of 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜from the oscilloscope trace (set the oscilloscope for a very slow sweep and use manual triggering. Compare with the rate calculated from the values of the resistors and the capacitor in the circuit. Change 𝑅𝑅1 and repeat your measurement. Do the two results agree with equation 3? Questions: As shown in Figure 8, use the function generator to apply a square-wave of frequency = 1kHz and 𝑉𝑉𝑝𝑝𝑒𝑒𝑒𝑒𝑝𝑝−𝑝𝑝𝑒𝑒𝑒𝑒𝑝𝑝 = 2V as 𝑉𝑉𝑖𝑖𝑖𝑖. Calculate the expected output signal 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜

-

+

_

VV

inout

R

C

pin 1 pin 8

20k pot.

+V

Figure 7. Voltage integrator circuit with drift control and small voltage applied to the input via a voltage divider (𝑅𝑅1 and 𝑅𝑅2) For simplicity, use the 10kΩ potentiometer on your Proto-Board. Remember to connect the power supply to the op-amp.

Switch

+V

R

R0

1

10kΩ potentiometer




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from equation 3 and compare to your experimental 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜. You will need to periodically reset the integrator by pushing the discharge button on the capacitor because the average voltage from the function generator is not exactly 0𝑉𝑉 and the drift compensation on your op-amp is not perfect. Repeat the derivation and comparison for a square-wave and a triangle wave at your three frequencies. You may either save the oscilloscope outputs to a file or take pictures with your cell phone. If you have time, repeat for a sine-wave input.

5 The Magnetic Field of the Earth We will now use the integrator in Section 3 to measure the magnetic field of the earth. The magnetic field of the earth varies in amplitude and direction with geographical position. A classical compass measures only the field direction in the 𝑥𝑥𝑥𝑥 direction. We will measure the full magnetic field vector in the laboratory. Build the circuit shown in Figures 9 and 10.

-

+

_

Vout

R

C

pin 1 pin 8

20k pot.

+V

Figure 8. Voltage integrator circuit with drift control and alternating voltage applied to the input. Remember to connect the power supply to the op-amp.

Switch

Function Generator




9

A large many-turn inductor coil is an excellent transducer for magnetic-field measurements because of Faradays law, which states that an electromotive force 𝜀𝜀 is induced in the coil when the magnetic field flux changes. When the coil is flipped by 180°, in a fixed magnetic field, the flux changes by twice the starting value. Thus, integrating the change in voltage suffices to determine the flux, according to:

𝑉𝑉𝑓𝑓𝑖𝑖𝑖𝑖𝑒𝑒𝑒𝑒 = −1𝑅𝑅𝑅𝑅

� 𝜀𝜀(𝑡𝑡′)𝑎𝑎𝑡𝑡′𝑜𝑜

0= −2

𝐴𝐴𝐴𝐴𝐴𝐴𝑅𝑅𝑅𝑅

, (4)

where 𝐴𝐴 is the component of the magnetic field in the direction of the coil axis, 𝐴𝐴 is the number of turns of the coil and 𝐴𝐴 the effective coil area. The average area of a multi-layer coil, whose mean radius is 𝑟𝑟 and whose maximum and minimum radii are 𝑟𝑟 ± 𝛿𝛿 , is:

𝐴𝐴 = 𝜋𝜋 �𝑟𝑟2 + 13𝛿𝛿2�. (5)

Choose the input resistor 𝑅𝑅 such that a single flip of the coil causes a 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜 that you can measure with at least 10% accuracy with the oscilloscope. Note that any drift in the integrator is faster when 𝑅𝑅 is smaller. You need not completely eliminate the drift; just make it small compared to the final value for 𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜. Make a series of measurements flipping the coil by 180°perpendicular to its axis. Repeat your measurement three times to measure: 𝐴𝐴𝑧𝑧 with the coil axis vertical, 𝐴𝐴𝑥𝑥 with N-S horizontal coil axis (along the lab room), and 𝐴𝐴𝑦𝑦 with horizontal E-W axis (perpendicular to both). Think carefully about which orientation

-

+

_

VV

inout

R

C

pin 1 pin 8

20k pot.

+V

Figure 9. Inductor connected to voltage integrator circuit with drift control and small voltage applied to the input via a voltage divider (𝑅𝑅1 and 𝑅𝑅2). Remember to connect the power supply to the op-amp.

Switch

Large Inductor Coil




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of the coil measures which axis of the earth’s magnetic field, which way you need to flip it, and include a sketch of the orientations and the rotations you performed in your lab book. For each orientation determine the amount of drift during the measurement and subtract it from your final values. Determine the component of the field in each direction. Combine the three components to get the orientation and magnitude of the B vector. The S.I. unit for 𝐴𝐴 (appropriate for equation 4) is 1T (Tesla). 1T = 104G. Questions: List possible sources for uncertainties. Evaluate the error of the three individual field measurements. Combine the errors to get the uncertainty of the magnitude 𝐴𝐴 of the field. Question: Compare your measurement of the earth’s magnetic field with the accepted value: http://ngdc.noaa.gov/seg/geomag/jsp/struts/calcPointIGRF.

Figure 10. Photo of the apparatus for the magnetic field measurement. The large coil is at the upper left. Most of them are mounted on gimbals to make them easier to rotate. Use the white plastic adjustment tool to set the 20kΩ potentiometer to minimize the drift in the integrator. Using a color scheme for the wires can help you keep track of the wiring. The 2uF capacitor has a push-button reset switch attached.

http://ngdc.noaa.gov/seg/geomag/jsp/struts/calcPointIGRF



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12 Last Revised by Mike Hosek, Sunny Nigam and James A. Glazier 9/20/15

lab #3: operational amplifiers - department of physics |...

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